Table Of ContentAsymmetric Spin-wave Dispersion on Fe(110): Direct Evidence of
Dzyaloshinskii–Moriya Interaction
Kh. Zakeri,1,∗ Y. Zhang,1 J. Prokop,1 T.-H. Chuang,1 N. Sakr,1 W. X. Tang,1,2 and J. Kirschner1
1Max-Planck-Institut fu¨r Mikrostrukturphysik, Weinberg 2, 06120 Halle, Germany
2School of Physics, Monash University, Victoria 3800, Australia
(Dated: January 23, 2017)
The influence of the Dzyaloshinskii–Moriya interaction on the spin-wave dispersion in an Fe
7 double-layer grown on W(110) is measured for the first time. It is demonstrated that the
1 Dzyaloshinskii–Moriya interaction breaks the degeneracy of spin-waves and leads to an asymmet-
0 ric spin-wave dispersion relation. An extended Heisenberg spin Hamiltonian is employed to obtain
2 thelongitudinalcomponentoftheDzyaloshinskii–Moriyavectorsfromtheexperimentallymeasured
n energy asymmetry.
a
J PACSnumbers: 75.30.Ds,75.70.Ak,75.70.Rf,75.50.Bb
0
2
In1957,Dzyaloshinskiiproposedanantisymmetricex- asymmetric. By measuring the highly resolved spin po-
] changeinteraction,basedonsymmetryarguments,toex- larizedelectronenergyloss (SPEEL)spectra in both en-
i
c plain the weak ferromagnetism observed in some oxide ergylossandgainregimesandbyreversingthe magneti-
s materialse.g. α−Fe O (Hematite)[1]. Onlythreeyears zationofthefilm,wemeasurethe DMinteractiondriven
- 2 3
l later it was shown by Moriya that, in principle, this in- asymmetryinthespin-wavedispersionofFedouble-layer
r
t teraction can be analytically derived by considering the grownonW(110). ByusinganextendedHeisenbergspin
m
relativistic spin-orbit correction in the Hamiltonian [2]. Hamiltonian, the energy asymmetry is modeled giving
. Theantisymmetricexchangeinteraction,Dzyaloshinskii– rise to a quantitative determination of the longitudinal
t
a
Moriya (DM) interaction, became very important to un- components of DM vectors.
m
derstand many physical properties of different systems The ironlayerwas deposited onto a clean W(110) sin-
- i.e. spin-glasses[3],cuprates[4],molecularmagnets[5,6] gle crystal at room temperature (RT). Special care has
d
n and multiferroics [7, 8]. been taken concerning the cleaning of the W crystal as
o In nanomagnetism, where the surface effects are no- described elsewhere [23]. Prior to the SPEELS measure-
c ticeable, the spin-orbit coupling is one of the most im- ments, the structural, chemical and magnetic properties
[
portant intrinsic magnetic perturbations, which creates werecheckedbymeansoflowenergyelectrondiffraction,
1 novel phenomena. Recently, it has been shown that a Auger electron spectroscopy and magneto optical Kerr
v strong spin-orbit coupling in the presence of the broken effect measurements. The Fe films reveal the expected
4
inversionsymmetryatthesurfaceleadstotheDMinter- structural and magnetic properties well-known from lit-
5
action, which stabilizes a noncollinear spin structure for erature [24–26]. The SPEELS measurements were per-
8
5 aMnmonolayeronW(110)[9]andW(100)[10]surfaces. formedusingourhighperformancespectrometerwithan
0 An ultrathin Fe film grown on W(110) is another sys- energy resolution well below 20 meV and a beam polar-
1. tem that is believedto showthe DM interaction[11–13]. ization of about 70±10%[27].
0 Magnetic excitations in this quasi-two-dimensional spin A schematic representation of the scattering geome-
7 system have been studied theoretically since many years try is given in the inset of Fig. 1(a). A well-defined
1
[14–21]. In the description of the collective magnetic ex- monochromatized spin-polarized electron beam is scat-
:
v citations, only the symmetric exchange interaction was tered from the sample and the electron energy loss and
i considered and the DM interaction has been neglected. gain spectra are measured as a function of the in-plane
X
Insuchsystems,whereDMinteractionisrelativelylarge, momentum transfer (∆Kk) for both spin orientations of
r
a it should, in principle, change the intrinsic properties of the incoming electron beam (up↑ and down↓). The sur-
the spin-waves (SWs). Only very recently, the influence faceSWs areexcitedinavirtualspin-flipscatteringpro-
of the DM interaction on the spin-wave dispersion has cess[28–31]. Theconservationoftheangularmomentum
been predicted to give rise to an asymmetric spin-wave duringthescatteringprohibitsSWexcitationsforincom-
dispersion in an Fe monolayeron W(110) [22]. However, ing electrons with a spin polarization antiparallel to the
the effectofthe DMinteractiononthe spin-wavedisper- sample magnetization(I ). Hence, only electronshaving
↑
sioninlowdimensionalsystemshasneverbeenmeasured minority spin character (I ) can create SWs. The elec-
↓
experimentally. trons with majority spin character(I ) can, in principle,
↑
InthisLetterwereportthefirstexperimentalevidence annihilate the thermally excited SWs while gaining en-
of the influence of DM interaction on the spin-wave dis- ergy. Thesefactsleadtoapeakintheminorityspinchan-
persion in a double-layer Fe. We show that in the pres- nel in the energy loss region and a peak in the majority
ence of the DM interaction the spin-wave dispersion is spinchannelinthe energygainregion(this is inanalogy
2
the peak positionsdue to the spin-wavecreationandan-
nihilation processes.
10k
(a) I + I In the absence of the DM interaction the spin-wave
s) [110] I dispersion has to be symmetric with respect to the en-
nts/ 8k I ergy axis, meaning that measuring the SW spectra for
u negative wave vector transfers has to result in the same
o
C 6k excitation energy as the one measured at positive wave
(
sity M [001] vector transfers; ∆E=E(∆Kk)-E(−∆Kk)=0.
n 4k Figure 2 shows a series of difference and asymmetry
e
nt curves measured on a 2 ML Fe film on W(110) at RT.
I
2k The full symbols are the results of measurements when
the magnetization is pointing along the [¯110]-direction.
Oneclearlyseesthatfor∆Kk=0.5˚A−1 the SWcreation
s) 2k (b) peak (energy loss) is at higher energies,whereas the SW
nts/ T=300K 20 %) annihilation peak (energy gain) is at lower ones (it can
Cou 1k y ( be seen better in the asymmetry curves). The situation
r
e ( 0 0 met
c
n m
ere-1k I - I sy
ff Asy.= -20 A
Di I+ I
-2k Diff. = I- I Energy gain Energy loss
-1
-3k Energy gain Energy loss -40 s) 3k (a) 0.5 ¯-1
-80 -40 0 40 80 120 nts/ 2k Diff. = I - I -0.5 ¯
u M|| [110]
Energy (meV) o
C 1k
(
FIG. 1: (color online). (a) SPEEL-spectra measured on a 2 e
MLFefilmepitaxially grownonW(110). Theincomingelec- nc 0
e
tronbeam hadan energyofE0=4eV.Insetshows thegeom- fer-1k
etry of ourSPEELS experiment. The spin-polarized electron Dif
beamisscatteredalongthe[001]-directionoftheFe(110)sur- -2k -1
face in the magnetic remanent state. The scattering angle is -0.5 ¯
kept at θ0 = 80◦. By changing the incident angle θ, the in- -3k 0.5 ¯-1
plane wave vector transfer parallel to the surface, ∆Kk, can
beadjusted(∆Kk =kfsin(θ0−θ)−kisin(θ),wherekiandkf 60 (b) M|| [110]
are the initial and final momentums of the electrons, respec- I - I
tively). For this experiment it was chosen to be ∆Kk=0.5 40 Asy.=
˚A−1. (b) The difference and asymmetry spectra. The big %) I + I
triangles showthepeak position duetotheSW creation and y ( 20
annihilationtakingplaceinenergylossandgain,respectively. etr
m
m 0
y
s
totheStokesandani-StokespeaksinaRaman/Brillouin A
-20
lightscatteringexperiment). Figure1(a)showsatypical
SPEEL spectra measured at ∆Kk=0.5 ˚A−1 on a 2 ML
-40
Fe film. The amplitude of the peak due to the SW anni-
hilation(in the energy gainregion)is much smallerthan
the one caused by the SW creation. This is due to the -80 -40 0 40 80 120
Energy (meV)
factthattheprobabilityofhavingthermallyexcitedSWs
in the system is given by the Boltzmann factor, which is
FIG. 2: (color online). (a) Series of difference, Diff.= I -I ,
↓ ↑
about 0.01–0.1 at RT. This gives rise to a large peak in and (b) asymmetry, Asy.=I↓−I↑, SPEEL spectra measured
the energy lossregionanda small dip in the energygain I↓+I↑
for ∆Kk=±0.5 ˚A−1 on a 2 ML Fe at RT. The filled symbols
region of the difference spectra. However, both features are for M k [¯110] and the open ones are for M k [1¯10]. The
can be seen clearly in the asymmetry curves, where the
Asy. = I↓−I↑ is plotted as a function of energy for both sapnecetnreargayrererseoclourtdioedn aotf a19bmeaemV.enTehregybiogftEr0ia=n4g.l1e6s3meaVrkwtihthe
I↓+I↑
loss and gain regions. In Fig. 1(b), the difference and peakpositionsofSWcreationsandannihilations,takingplace
asymmetrycurvesarepresented. The big trianglesmark at energy loss and gain, respectively.
3
is totally reversed for negative wave-vector transfers i.e.
∆Kk=-0.5 ˚A−1; the SW annihilation peak is at higher 12 (a) Theo.: c=+1,ê||[110]
energies and SW creation peak is at lower energies now.
c=+1,ê||[110]
Ifthis effect iscausedbyanuncertaintyinthe wavevec- ) 8
V
tortransfer,duetothestrayfieldsinducedbendingofthe e
m
electronbeamintwodifferentexperiments,onewouldex- ( 4
||K
pectthesameeffectinthegainandlossregions(increase D
or decrease in both energies). The reversed phenomena E-0
-
in energy gain and loss regions indicate that this effect ||K
D
E-4
cannot be due to a slightly different electron trajectory
=
in two different experiments. E
Another argument, which clarifies that this is an intrin- D -8 { M||[110]
Expt.:
sic property of the system comes from measuring the M||[110]
-12
same spectra for opposite magnetization directions. In
-1.8 -1.2 -0.6 0.0 0.6 1.2 1.8
magnetism,reversingthe sample magnetizationis equiv- WavevectorDK||(Å-1)
alent to time inversion (in our experiment it basically
means that the beam source and the detector are inter- (b) N: Nearest neighbor
changed). The data for magnetization along the [1¯10]- NN Rij=Ri-Rj in the same atomic plane
direction are shown by open symbols in Fig . 2. In N N’ N N’: Nearest neighbor
the case of reversed magnetization the SW excitation in the neighboring atomic plane
peak for ∆Kk=-0.5 ˚A−1 is at higher energies with re- N’N’ R N’N’
ij NN: Next nearest neighbor
spect to the one for ∆Kk=0.5 ˚A−1. This clearly indi-
cates that having a slightly different energy for the SWs ]10 N N’ N in the same atomic plane
0
propagatingalongthe [001]-directionwithrespectto the [ NN N’N’: Next nearest neighbor
ones propagating along the [00¯1]-direction is an intrinsic [110] in the neighboring atomic plane
property of the SWs in this particular system. Based
on the spin wave theory the symmetric exchange inter- FIG.3: (coloronline)(a)Theenergyasymmetryasafunction
action cannot lead to any degeneracy breaking of the ofwavevectortransfer. Thesymbolsarethemeasuredvalues
spin waves. One may think about presence of the dipo- fortwodifferentmagnetizationdirectionsandthesolidcurves
arethefitsusingEq. (2)forc=+1anddifferentmagnetiza-
lar interaction that is responsible for the unidirectional
tion directions. The error bars represent both the statistical
Damon-Eshbach surface modes [32]. In this case the en-
and systematic uncertainties. (b) Real space representation
ergy difference should be about 0.1 meV, which is much
of a 2 ML Fe slab.
smaller than values observedin our experiment. Finally,
we conclude that the presence of DM interaction breaks
the degeneracy of the SW energies and leads to differ- DM vectors from our experimental data. By starting
with a simple classical description of the SWs, the mod-
entenergiesfor a given∆K . Therefore,the assumption
k
ified Heisenberg Hamiltonian in the presence of the DM
∆E(∆K )=∆E(−∆K ) is not valid here anymore.
k k
interaction can be written as: H = J S ·S −
It is worth to mention that measurements on a 20 ML i6=j ij i j
thicksampleshowedalsoanenergyasymmetry. Theval- Keff i(Si·eˆ)2+ i6=jDij·Si×Sj. HPerethefirstterm
uesoftheenergyasymmetryinthiscasearesmallerthan is thePsymmetric Pexchange interaction (Jij is the ex-
theonesmeasuredforthedouble-layer. Thisobservation change coupling constant between spins Si and Sj), the
reveals two facts: (i) since SPEELS is only sensitive to second term is the magnetic anisotropy energy (MAE)
the top most layer(s), this effect is more likely a surface term (Keff is the effective MAE constant with an easy
effect and is preservedup to even 20 ML thick films, (ii) axis along eˆ) and the last term is the DM interaction
this effect has nothing to do with the stray fields caused term(Dij aretheDMvectors). Thelasttermistheonly
by the sample, because the stray fields strength is pro- one, which leads to an asymmetric dispersion relation.
portional the film thickness. If this effect was caused Using the same notation as in Ref. [22], the asymmetry
by stray fields, one would expect a larger effect for the in the SW energies, ∆E =EDM(q)−EDM(−q), reads:
thicker films.
The energy asymmetry, ∆E = E(∆Kk)−E(−∆Kk),
∆E = 2csin2θ (D ·eˆ)sin[q·(R −R )], (1)
inducedbyDMinteractionvarieswiththe in-planewave ij i j
Xi6=j
vector transfer. In Fig. 3(a) the energy asymmetry is
plottedasafunctionofthein-planewavevectortransfer, where q is the wave vector of the SWs (in our case
∆Kk. Our data show that ∆E has a distinct maximum SWs are propagating along the [001]-direction,therefore
at ±0.5<∆Kk <±1. |q| =∆Kk), c=±1 is the chirality rotation index (being
Now, we attempt to estimate the amplitude of the +1 for right rotating sense and -1 for the left rotating
4
one), eˆ is the unit-vector of the magnetization M, θ is
the relativeanglebetweenmomentsandeˆandR (R )is
i j
the position vector of site i(j). For a double-layer slab
∗ Corresponding author: [email protected]
Eq. (1) can be simplified to:
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